CHAPTER 15: BARGAINING Multiagent Systems http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e Overview • How do agents reach agreements when they are self interested? • In an extreme case (zero sum encounter) no agreement is possible — but in most scenarios, there is potential for mutually beneficial agreement on matters of common interest . 1 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e Overview • The capabilities of: – negotiation and – argumentation are central to the ability of an agent to reach such agreements. 2 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e Two pictures that summarise negotiation 3 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e Mechanisms, Protocols, and Strategies • Negotiation is governed by a particular mechanism , or protocol . • The mechanism defines the “rules of encounter” between agents. • Mechanism design is designing mechanisms so that they have certain desirable properties. – Properties like Pareto efficiency • Given a particular protocol, how can a particular strategy be designed that individual agents can use? 4 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e Auctions versus Negotiation • Auctions are only concerned with the allocation of goods: richer techniques for reaching agreements are required. • Negotiation is the process of reaching agreements on matters of common interest. 5 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • Any negotiation setting will have four components: – A negotiation set: possible proposals that agents can make. – A protocol. – Strategies, one for each agent, which are private. – A rule that determines when a deal has been struck and what the agreement deal is. Negotiation often proceeds in a series of rounds, with proposals at every round. 6 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • There are a number of aspects of negotiation that make it complex. • Multiple issues – Number of possible deals is exponential in the number of issues. (Like the number of bundles in a combinatorial auction) – Hard to compare offers across multiple issues The car salesman problem • Multiple agents – One-to-one negotiation 7 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e – Many-to-one negotiation – Many-to-many negotiation • At the simple end there isn’t much to distinguish negotiation from auctions. 8 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e Negotiation for Resource Division • We will start by looking at Rubinstein’s alternating offers model. • This is a one-to-one protocol. • Agents are 1 and 2 , and they negotiate over a series of rounds: 0 , 1 , 2 , . . . • In round 0 , Agent 1 makes an offer x 0 . • Agent 2 either accepts A , or rejects R . • If the offer is accepted, then the deal is implemented. • If not, we have round 1 , and Agent 2 makes an offer. 9 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e start Agent 1 makes a proposal Agent 2 accepts Agent 2 rejects Agent 1 rejects Agent 2 makes a proposal 10 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • The rules of the protocol don’t mean that agreement will ever be reached. – Agents could just keep rejecting offers. • If there is no agreement, we say the result is the conflict deal Θ . • We make the following basic assumptions: – Disagreement is the worst ouctome Both agents prefer any agreement to none. – Agents seek to maximise utility Agents prefer to get larger utility values • With this basic model, we get some odd results. 11 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • Consider we are dividing a pie . . . 12 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • Model this as some resource with value 1, that is divided into two parts. – Each part is between 0 and 1. – The two parts sum to 1 so a proposal is ( x , 1 − x ) • The set of possible deals is: { ( x , 1 − x ) : 0 ≤ x ≤ 1 } • If you are Agent 1, what do you offer? 13 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • Let’s assume that we will only have one round. Ultimatum game • Agent 1 has all the power. • If Agent 1 proposes (1 , 0) , then this is still better for Agent 2 than the conflict deal. • Agent 1 can do no better than this either. • So we have a Nash equilibrium. 14 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • If we have two rounds, the power passes to Agent 2. • Whatever Agent 1 proposes, Agent 2 rejects it. • Then Agent 2 proposes (0 , 1) . • Just as before this is still better for Agent 1 than the conflict deal and so it is accepted. • A bit of thought shows that this will happen any time there is a fixed number of rounds. 15 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • What if we have an indefinite number of rounds. • Let’s say that Agent 1 uses this strategy: Always propose (1 , 0) and always reject any offer from Agent 2 • How should Agent 2 respond? • If she rejects, then there will never be agreement. – Conflict deal • So accept. And there is no point in not accepting on the first round. 16 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • In fact, whatever ( x , 1 − x ) agent 1 proposes here, immediate acceptance is the Nash equilibrium so long as Agent 2 knows what Agent 1’s strategy is. 17 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e Impatient players • Since we have an infinite number of Nash equilibria, the solution concept of NE is too weak to help us. • Can get unqiue results if we take time into account. For any outcome x and times t 2 > t 1 , both agents prefer x at time t 1 . • A standard way to model this impatience is to discount the value of the outcome. • Each agent has δ i , i ∈ { 1 , 2 } , where 0 ≤ δ < 1 . • The closer δ i is to 1, the more patient the agent is. 18 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • If agent i is offered x , then the value of the slice is: – x at time 0 – δ i x at time 1 – δ 2 i x at time 2. . . . – δ k x at time k • Now we can make some progress with the fixed number of rounds. • A 1 round game is still an ultimatum game. 19 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • A 2 round game means Agent 2 can play as before, but if so, will only get δ 2 . Gets the whole pie, but it is worth less. 20 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • Agent 1 can take this into account. • If Agent 1 offers: (1 − δ 2 , δ 2 ) then Agent 2 might as well accept — can do no better. • So this is now a Nash equilibrium. 21 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • In the general case, agent 1 makes the proposal that gives Agent 2 what Agent 2 would be able to enforce in the second round. • Agent 1 gets: 1 − δ 2 1 − δ 1 δ 2 • Agent 2 gets: δ 2 (1 − δ 1 ) 1 − δ 1 δ 2 • Note that the more patient either agent is, the more pie they get. 22 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e Heuristic approach • The approach we just talked about relies on strageic thinking about the other player. • A simpler approach is to use some heuristic approximation of how the value of the pie varies for the players. • Some common approximations: – Linear – Boulware – Conceder • We can see what these look like for buyers. 23 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e 24 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
Chapter 15 An Introduction to Multiagent Systems 2e • Linear – Linear increase from initial price at the start time to reserve price at the deadline. • Boulware – Very slow increase until close to deadline and then an exponential increase. • Conceder – Inital exponential increase to close to the reserve price and then not much change. 25 http://www.csc.liv.ac.uk/˜mjw/pubs/imas/
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